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Thus a function can be constructed, which takes values from the co-domain of back to the circle using the inverse, followed by back to the interval. If a is any number in , then:

Such a function is called a ''transition map''.

The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts

and

Here s is the slope of the line through the point

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